Godley began his acting career at the age of 9, in a BBC radio production of Hemingway's My Old Man. His first stage role came at age 11, as Prince Giovanni in The White Devil at the Old Vic Theatre. His childhood career also included work at the National Theatre, in Lillian Helman's Watch on the Rhine, and Close of Play, directed by Harold Pinter.

After playing the lead in the BBC TV adaptation of J. Meade Faulkner's Moonfleet, Godley joined Alan Ayckbourne's theatre company in Scarborough, where he stayed for three seasons. Highlights included June Moon and The Revengers' Comedies, both of which transferred to the West End, and Mr A's Amazing Maze Plays, which transferred to the National Theatre.

He spent one season as a member of the Royal Shakespeare Company, before creating the role of Cliff in Sam Mendes's production of Cabaret at the Donmar Warehouse. Several major productions followed, including Mouth to Mouth at the Royal Court, The Front Page at the Donmar, after which Godley went on to create a series of roles at the National, including Kenneth Williams in Cleo, Camping, Emmanuelle and Dick; Michal, in Martin McDonagh's The Pillowman, and the title role in Howard Davies' production of Paul. He also played Jack Worthing in the West End production of The Importance of Being Earnest and Victor in both the West End and 2001 Broadway production of Noël Coward's Private Lives. He was most recently seen as Lord Evelyn Oakleigh in the Broadway revival of Anything Goes, and as Raymond Babitt in the West End production of Rain Man opposite Josh Hartnett.

In addition to his stage work, Godley has also appeared in numerous films, including Battleship, Wilde Salome, The X-Files: I Want To Believe, Elizabeth: The Golden Age, Son of Rambow, Nanny McPhee, Charlie and the Chocolate Factory, Around the World in 80 Days, and Love Actually.

His television roles include Mad Men, Breaking Bad, and The Good Wife, and the mini-series Nuremberg and A Young Doctor's Notebook. TV films include Cor, Blimey!, The Old Curiosity Shop, The Young Visiters, Hawking, and the HBO film The Special Relationship.